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Let's break down the problem step by step.

### Problem Restatement:
Jo uses:
- 1 bread roll
- 2 cheese slices

to make 1 cheese sandwich.

She needs to:
1. Buy enough packs so that she has **exactly twice as many cheese slices as bread rolls**.
2. Make **more than 100 cheese sandwiches**.

The packs available are:
- Pack of 15 bread rolls for £1.83.
- Pack of 25 cheese slices for £2.24.

### Step 1: Determine the Number of Packs Needed
Let:
- \( x \) be the number of bread roll packs.
- \( y \) be the number of cheese slice packs.

Each bread roll pack contains 15 rolls, so:
\[ \text{Total bread rolls} = 15x \]

Each cheese slice pack contains 25 slices, so:
\[ \text{Total cheese slices} = 25y \]

We know:
\[ \text{Total cheese slices} = 2 \times \text{Total bread rolls} \]
\[ 25y = 2 \times 15x \]
\[ 25y = 30x \]

This simplifies to:
\[ y = \frac{6x}{5} \]

Since \( y \) must be an integer, \( x \) must be a multiple of 5.

### Step 2: Ensure More Than 100 Sandwiches
One sandwich requires 1 bread roll and 2 cheese slices. Therefore, the number of sandwiches Jo can make is the same as the number of bread rolls.

\[ 15x > 100 \]
\[ x > \frac{100}{15} \approx 6.67 \]

So \( x \geq 7 \).

### Step 3: Calculate \( y \) for Each Possible \( x \) Value
Given \( x = 5k \) for some integer \( k \), let's test \( x = 10 \) (the smallest multiple of 5 greater than 6.67).

For \( x = 10 \):
\[ y = \frac{6 \times 10}{5} = 12 \]

### Step 4: Cost Calculation
Now calculate the total cost:
- Cost of bread rolls: \( 10 \times 1.83 = £18.30 \)
- Cost of cheese slices: \( 12 \times 2.24 = £26.88 \)

Total cost:
\[ 18.30 + 26.88 = £45.18 \]

Thus, the least amount Jo can spend to meet the conditions is **£45.18**.

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Would you like further details or have any questions?

Here are 5 related questions to further your understanding:
1. How would the cost change if the number of sandwiches needed was exactly 100?
2. What if the pack sizes were different, say 20 bread rolls and 30 cheese slices?
3. Could you determine the cost for making fewer sandwiches, say 50 or 75, using the same method?
4. How would the equation change if Jo needed a 1:1 ratio of cheese slices to bread rolls?
5. What if Jo wanted to minimize waste instead of cost?

**Tip:** When working on optimization problems like this, always check if small adjustments in pack quantities (like using the smallest multiple of packs) could meet the conditions with the least cost.

To make one cheese sandwich, Jo uses one bread roll and two cheese slices. She needs to buy packs of bread rolls (15 per pack) and cheese slices (25 per pack) at £1.83 and £2.24 respectively, ensuring she has exactly twice as many cheese slices as bread rolls and can make more than 100 sandwiches. What is the least amount she can spend?

Learn how to minimize cost while making more than 100 cheese sandwiches using the right ratio of cheese slices to bread rolls. This problem covers algebraic optimization, ratio and proportion, and rational equations. Suitable for students in Grades 9-12.

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